# Analyse the Kinetic Energy Equation

In this worksheet, students will rearrange the kinetic energy equation to solve complex problems.

### QUESTION 1 of 10

What would you rather be hit by, a speeding feather or a speeding train?

It’s the feather, right? But why? We'll be looking at kinetic energy in this activity and by the end of it, you should be able to remember, use and rearrange the equation for kinetic energy to solve complex problems.

(A heads up - this activity will assume you know how to rearrange equations and have a good mathematical understanding of applying them).

Are you ready?  Let’s get stuck in!

As you should already know, kinetic energy is how moving things store their energy. There are a few things that affect this and we'll be using the idea of the feather and the train to try and understand the different factors that affect kinetic energy.

Why would you rather be hit by a feather? Because it’s light, right? This is true, the mass of the object will affect how much kinetic energy it has. The more massive an object, the more energy it will have.  Let’s make a note of this.

Energy (E) ∝ mass (m)

That fancy sign in the above equation (∝), is called the proportional sign. When something is proportional, it means that if one thing goes up, then the other must go up as well. Here, this means that if we change the mass, we also change the energy.

Let’s change the situation a little bit now. The train is moving at 0.01 m/s and the feather is moving at 1,000 m/s. Which one would you rather be hit by now? Did you go for the train? That’s because speed (or velocity) also has an effect on the kinetic energy (if you want to know more on this and why this works,  have a look at our activity on momentum - it will explain this in a lot more detail). Let’s add this to our proportionality:

Energy (E) ∝ mass (m) x velocity (v)

All we're saying here is that if we change either the mass or the velocity, then it will affect how much energy there is. If either of them goes up, then the energy will also go up - the same applies if they go down.

Oh – but it isn’t as simple as that… scientists have conducted experiments and found that one of these things has a much more significant effect on energy than the other. When we draw graphs of energy against mass and velocity, we find that the mass graph forms a line of about y = ½ x and the velocity graph looks like a y = x² graph!

By using these two graphs, we know that the mass has to be halved  and the velocity needs to be squared.  This leads us to the equation - we have to halve the mass and square the velocity, so the equation looks like this:

Fun fact time! You have to remember this equation - you will not be given it in the exam.

So how do you use this equation? Simple! Let’s go through an example together - this is where the maths comes into play!

Jim is an expert at throwing. He can literally throw anything (including a tantrum) using a specific amount of energy - it is always 500 J. One day, he throws a ball at a speed of  5 m/s. Calculate the mass of the ball.

Step 1   Find the values that matter to you in this question and highlight them:

Jim is an expert at throwing. He can literally throw anything (including a tantrum) using a specific amount of energy - it is always 500 J. One day, he throws a ball at 5 m/s. Calculate the mass of the ball.

Step 2  Work out if you need to rearrange the equation:

In this example, we need to rearrange for mass.

The equation is currently E = ½mv²

Firstly, multiply out the ½

Then, divide by the v²

This makes mass the subject of the equation: m = 2E ÷ v²

Step 3  Put these values into the equation:

m = (2 x 500) ÷ 5²

Step 4  Put these into your calculator and press =

m = 40 kg (it's a heavy ball)

Step 5  Round if you need to. Normally we round to 1 decimal place (dp) or 2 significant figures (sf) unless the question tells you otherwise!

No rounding is needed in this answer.

Don’t forget the unit of energy is the joule (J)!

Now let's try some questions.

What unit is used for energy?

A lift has a maximum capacity of 12 people and can travel at 14 m/s.

The lift has a mass of 250 kg and a person has a mass of approximately 75 kg.

If the lift is half full, what is its kinetic energy?

A car's brakes when fully depressed can take 12,000 J of energy out of the car for every metre of braking distance.

A car on the motorway has a mass of 1,500 kg and is travelling at 30 m/s.

Calculate the distance it takes it to stop, once the brakes are depressed.

Both the velocity and the mass of the vehicle will make a difference to braking distances.

Explain which will have a bigger impact.

Use your knowledge from the last question to answer this one and assume that all brakes will take out the same amount of energy.

[3]

Calculate the velocity of an object that has a mass of 2 kg, and has gained 250 J of gravitational energy when it hit the floor?

Assume that there is no energy lost to the surroundings.

Calculate the mass of an object that has 104,040 J of energy with a velocity of 6 m/s.

Sandra is riding a bike with an energy of 1224 J and a velocity of 6 m/s.

If she has a mass of 30 kg, what is the mass of the bike?

A lift can have a maximum energy of 68,600 J at an operating speed of 14 m/s. Each person has a mass of 75 kg, and the mass of the lift is 250 kg.

Calculate the total number of people that can use the lift at the same time.

Your younger brother has let go of his balloon. As you watch it moving higher and higher up into the sky, you think "I wonder how fast that balloon is travelling."  Then you realise you can work it out!  Let's say this balloon has a mass of 0.05 kg and it'll have an energy of about 268 J.

You break out your paper and calculator and work out that the balloon is moving at...

Calculate the speed of an object that has lost 478 kJ of energy from falling. Assume that no energy has been lost to the environment.

The mass of the object is 120 kg.

• Question 1

What unit is used for energy?

joule
joules
J
EDDIE SAYS
Named after the scientist who first defined it, the unit of energy is called the joule. You could have said 'joule' or just simply 'J' as they are both used to represent the unit of energy. In equations, however, we tend to use J instead of writing the whole thing out.
• Question 2

A lift has a maximum capacity of 12 people and can travel at 14 m/s.

The lift has a mass of 250 kg and a person has a mass of approximately 75 kg.

If the lift is half full, what is its kinetic energy?

EDDIE SAYS
A gentle start for you with this one! No rearrangement is needed to the equation in this question, as it's asking for the kinetic energy to be calculated. There are a number of things that need to be done to the mass before we start our calculations: 1 The lift is half full, so 12 ÷ 2 = 6. 2 Each person weighs 75 kg, so 75 x 6 = 450 kg. 3 The mass of the car is 250 kg, so we need to add that on. That makes the final mass 700 kg. Once we have done that, it is time to put it into the equation: E = (0.5 x 700) x 142 E = 68,600 J Simple isn't it?
• Question 3

A car's brakes when fully depressed can take 12,000 J of energy out of the car for every metre of braking distance.

A car on the motorway has a mass of 1,500 kg and is travelling at 30 m/s.

Calculate the distance it takes it to stop, once the brakes are depressed.

EDDIE SAYS
For this question, you need to work out the amount of energy in the car to begin with. m = 1500 kg v = 30 m/s E = 700 x 900 E = 675,000 J Okay, now we need to work out how far something is going to travel if 12,000 J of energy are taken out every metre. This is simple, you just divide 675,000 by 12,000 and you get your answer: 56.25 m.
• Question 4

Both the velocity and the mass of the vehicle will make a difference to braking distances.

Explain which will have a bigger impact.

Use your knowledge from the last question to answer this one and assume that all brakes will take out the same amount of energy.

[3]

EDDIE SAYS
Don't worry if you found this one a bit tricky. Questions that require you to write an explanation are always a bit harder, but you will find they become easier with practice. You must refer to the equation to get the full marks. This is the crux of questions like this - they want you to demonstrate that you understand the maths behind the equation. So, a good answer to this one would be on the lines of this: Velocity will make the most difference because it is squared in the equation. Mass will not make as much of a difference because it is halved in the equation. You could also get full marks for talking about proportionality - for example: Energy is proportional to velocity squared. Energy is proportional to half of the mass.
• Question 5

Calculate the velocity of an object that has a mass of 2 kg, and has gained 250 J of gravitational energy when it hit the floor?

Assume that there is no energy lost to the surroundings.

EDDIE SAYS
For this question, you need to rearrange the equation to put velocity as the subject. You should have an equation that looks like this: v = √(2E ÷ m) Then it is just a case of plugging in the numbers: E = 250 J m = 2 kg (2 x 250) ÷ 2 = 250 √250 = 15.8 m/s once you've rounded it to 1 dp. So, this is about as hard as it gets (for kinetic energy)! Always remember to write the calculation down in order to work it out. It's a tricky one, so don't worry if this tripped you up a bit.
• Question 6

Calculate the mass of an object that has 104,040 J of energy with a velocity of 6 m/s.

EDDIE SAYS
Okay, can you remember the equation for mass? It is: m = 2E ÷ v² Now you are left with the numbers: E = 104040 J v = 6 m/s Put these numbers into the equation: (2 x 104040) ÷ 62 m = 5780 kg
• Question 7

Sandra is riding a bike with an energy of 1224 J and a velocity of 6 m/s.

If she has a mass of 30 kg, what is the mass of the bike?

EDDIE SAYS
Quite a lot to do here as this would be a minimum of 3 marks in an exam. Definitely worth knowing how to work this one out, then! First of all, you would get a mark for rearranging the equation correctly. m = 2E ÷ v² Secondly, you would get a mark for calculating the total mass: m = (2 x 1224) ÷ 62 m = 68 kg Finally, you would get a mark for subtracting Sandra's mass from the total mass: m = 68 - 30 m = 38 kg What a question! Well done if you got it right!
• Question 8

A lift can have a maximum energy of 68,600 J at an operating speed of 14 m/s. Each person has a mass of 75 kg, and the mass of the lift is 250 kg.

Calculate the total number of people that can use the lift at the same time.

6
EDDIE SAYS
Okay, this is a question and a half! Let's break it down... Step 1 It's asking us to work out a mass, so rearrange the equation to get this: m = 2E ÷ v² Step 2 Plug in the numbers to get a total mass: m = (2 x 68,600) ÷ 142 m = 700 kg Step 3 Take away the mass of the lift itself. m = 700 - 250 m = 450 kg Step 4 Divide this by the mass of each person to find the number of people. 450 ÷ 75 = 6 This means there are 6 people in the lift. Phew... when you look at it like this, it's not as bad as it first seems, so don't panic - just take it one step at a time!
• Question 9

Your younger brother has let go of his balloon. As you watch it moving higher and higher up into the sky, you think "I wonder how fast that balloon is travelling."  Then you realise you can work it out!  Let's say this balloon has a mass of 0.05 kg and it'll have an energy of about 268 J.

You break out your paper and calculator and work out that the balloon is moving at...

EDDIE SAYS
Nearly there, only one question after this one and then you can take a well-earned break! The same format as always though, rearrange the equation first to get this: v = √ (2E ÷ m) Then plug in the numbers: v = √ ((2 x 268) ÷ 0.05) v = √ 10720 v = 103.5 m/s REMEMBER to round your answer to 1 dp.
• Question 10

Calculate the speed of an object that has lost 478 kJ of energy from falling. Assume that no energy has been lost to the environment.

The mass of the object is 120 kg.

EDDIE SAYS
Last one, but it does look just a bit complicated! In fact, the only real complication here is the significant figures part of the question. You're only allowed to put 2 significant figures, so if the number is bigger than 10 that means no decimal places at all! Everything else is using the same format that we have used for all of the other questions: Rearrange the equation first to get this: v = √ (2E ÷ m) Then plug in the numbers. Be careful - the energy is in kJ, so convert it into J before you put it into the equation, by multiplying it by 1000. v = √ (( 2 x 478,000) ÷ 120) v = 88.9758............... REMEMBER to round your answer to 2 significant figures. In reality, you wouldn't get anything this hard in an exam - but it's nice to know you can do it. You'll find all of the questions a lot easier now! Well done you!